**The point of contact of tangent** with the circle at point S and T. then each of the length **PS and PT** is called **tangential distance** from P to the circle. More explanation,

Suppose that we have any point outside the circle** p(x₁, y₁).** Then equation of form is

**x²+y²+2gx+2fy+c =0…………………….. A**

we know that two real and **distinct distance** can be drawn to the circle from an **external point**. If the point of contact of these tangent with the circle are S and T. then the **length PS and PT** is called the **length of the tangent** or tangential distance from P to the circle A.

The center of the circle has coordinated (-g, -f) joint PO and OT. From right triangle OTP

Tangential distance =

=

We can also find the tangential distance PS from this formula.

**Tangential distance example:**

**EXAMPLE :1**

Fined the length of the tangential distance from the point P(-5, 10) to the circle

**5x²+5y²+14x+12y-10=0**

**solution: **

Equation of the given circle in standard form is

dividing both side of equation by 5

**x²+y²+14/5 x+12/5y-2=0……………(A)**

Square the length of the tangent from point P (-5, 10) to the circle. A is obtained by substituting

**X = -5**

**y= 10**

we get length of the tangential distance

tangential distance =

**CORD OF CONTACT:**

**Example:**

Tangent are drawn from **(-3. 4) to the circle x² + y² = 21**. We fined an equation of the line

joining the point of contact. This line is called the **chord of contact.**

**Solution:**

let the point of contact of two tangent be P(x1, y1) and Q (x2, y2)

An equation of tangent at point P is

**XX1+yy1= 21………………….a**

An equation of tangent at point Q is

**XX2+yy2 = 21 ……………………b**

since (1) and (2) pass through (-3,4), so

**3×1+4y1=21………………. c**

and

-3×2+4y2=21………………’d

(c) and (d) show that the both point

** P(x1, y1), Q(x2, y2) lies on -3x+4y=21**

and so it is the required equation of chord of contact.

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